?? so .... given pre-stack of categories, consider ... ?? "quasicoherent presheaves of presheaves" over (?? ...) it ... ??? and whether (?? or in what generality ...) equivalent to filteredly cocontinuous presheaves on lax colimit ... ???? ....

?? "constant" case as maybe good case to experiment with ??? ...

?? then ... ?? "generalized day convolution" .... adapted to filteredly cocontinuous case .... ???? .....